Quantum computing differs from conventional computing because instead of storing and manipulating data in the form of binary bits, it uses quantum bits, which are commonly referred to as “qubits”. Unlike a binary bit, which can exist in only one of two discrete states, a qubit can exist in any quantum superposition of its two qubit states. Likewise, algorithms are implemented in conventional computing by binary logic gates, but in quantum computing they are implemented using quantum gates that operate on the quantum superposition and that are read out by collapsing the final quantum superposition to a single, measurable state that is detectable by, e.g., electrical or optical probes.
Quantum computing has been a research subject of intense interest over the past few years because of its potential for solving problems in cryptography, simulation, and other areas that would be extremely costly, or even intractable, by conventional computation. Some of the research has been directed to the search for a physical implementation of a qubit that can maintain a quantum state for a sufficient coherence time to permit computational operations to take place. Of course, any such implementation must include an isolated particle or excitation that can be manipulated between two distinct quantum states.
For example, particle-based approaches have used the internal states of trapped ions or neutral atoms, the spin of confined electrons, and even nuclear spins of dissolved molecules and dopant ions in solid state materials. Other examples come from the field of superconducting quantum computing. In those examples, the different quantum states might be, e.g., different numbers of Cooper pairs on a superconducting island, different numbers of magnetic flux quanta trapped in a superconducting ring, or different amplitudes of charge oscillation across a Josephson junction.
Approaches based on confined electrons are especially interesting because as a spin-½ particle, the electron inherently has precisely two mutually orthogonal spin states (often referred to as “spin up” and “spin down”). Moreover, the technology is available for electrostatically isolating individual electrons, for manipulating spin states using electric and/or magnetic fields, and for detecting the presence of confined electrons in individual quantum dots. (For reasons to be explained below, manipulations that change the quantum state of the confined electrons are referred to as “state rotations”.)
In 1998, for example, Daniel Loss and David P. DiVincenzo proposed one-qubit and two-qubit gates using the spin states of coupled single-electron quantum dots. Logical operations would be performed through gating of a tunneling barrier between neighboring dots. Their proposal is described in D. Loss et al., “Quantum computation with quantum dots,” Phys. Rev. A 57 (Jan. 1, 1998A) 120-126, the entirety of which is hereby incorporated herein by reference.
Various physical implementations of the ideas of Loss and DiVincenzo have been attempted since the publication of their article. One such implementation is the lateral gate quantum dot, or, simply, the “lateral dot” as discussed, for example, in R. Hanson et al., “Spins in few-electron quantum dots,” Rev. Mod. Physics 79 (October-December 2007) 1217-1265, the entirety of which is hereby incorporated herein by reference.
In one example provided by Hanson et al., a two-dimensional electron gas (2 DEG) forms at the interface between a gallium arsenide (GaAs) substrate and an overlayer of aluminum gallium arsenide (AlGaAs) grown by molecular beam epitaxy (MBE). Metal gate electrodes are deposited over the GaAs—AlGaAs heterostructure and patterned by electron-beam lithography. Negative voltages on the gate electrodes locally deplete the 2 DEG and in that way can be used to isolate small islands of electrons, thereby defining the quantum dots. 2 DEG reservoirs in the patterned structure are provided with ohmic contacts. The device is operated at cryogenic temperatures, typically 20 mK.
One of the examples that Hanson et al. provides is the double quantum dot. In a double quantum dot, electrons can be transferred through a tunnel barrier from one dot to the other by changing the electrostatic potentials applied through the gate electrodes to the respective dots. (When both dots are at the same potential, they are said to be “in resonance”. When they are out of resonance, they are said to be “detuned”. The amount of detuning is described by a potential difference ε.) However, some interdot charge transitions, i.e., some transfers of an electron from one dot to the other are forbidden by spin selection rules.
The charge states (i.e., the occupancy types) of the double dot that are of interest are (0,1), (1,1), and (0,2), in which, respectively, one dot is empty and one dot is occupied by a single electron, each dot is occupied by a single electron, and one dot is empty and the other dot is occupied by two electrons.
The two-electron ground state is the singlet state. For the (0,2) charge state, it is represented byS(0,2)=(|↑2↓2−|↓2↑2)/√{square root over (2)},where the arrow direction indicates “spin up” or “spin down” respectively, and the subscript indicates that the electron resides in dot i, where i=2. For the (1,1) charge state, the singlet spin state is represented byS(1,1)=(|↑1↓2−|↓1↑2)/√{square root over (2)}.
The stationary states next higher in energy are the manifold of three triplet states. However, a magnetic field applied at sufficient strength will split the triplet states, and it will effectively remove the T+ and T− (m=1) triplet states from consideration. The remaining triplet state is the T0 state. For the (0,2) charge state, the remaining triplet is represented byT0(0,2)=(|↑2↓2+|↓2↑2)/√{square root over (2)}.For the (1,1) charge state, it is represented by:T0(1,1)=(|↑1↓2+|↓1↑2)/√{square root over (2)}.
The so-called “exchange energy” is the energy difference J between the S and T0 states. The exchange energy is very strongly affected by hybridization of the electron wavefunctions when electrons are partially transferred from one dot to the other. As a consequence, the exchange energy is very sensitive to the detuning, ε.
Hanson et al. explains how the detuning can be used to perform state rotations on the confined electrons. The possible pure quantum states |Ψ(θ,φ that are linear combinations of the S and T0 states can be expressed in the form|Ψ(θ,φ=cos(θ/2)|S+exp(iφ)sin(θ/2)|T0,where the polar angle θ varies from 0° to 180° and the azimuthal angle φ varies from 0° to 360°.
Two of the possible states that can be reached with suitable values of θ and φ, are |↑↓ and |↑↓. (The charge state is neglected here to simplify the discussion.) The exchange energy J(ε) mixes these states, so that an initial |↑↓ state will evolve into a |↓↑ state in a time τ given by the expression τ=πℏ/J(ε). (The symbol ℏ represent s Planck's constant.)
The two states are related by a difference of 180° between their respective values of φ. In fact, these states can be represented as antipodal points on the surface of a sphere, referred to as the “Bloch sphere”. Each point on the surface of the Bloch sphere represents one of the states |Ψ(θ, φ referred to above. Accordingly, the evolution of the |↑↓ into the |↓↑ state may be regarded as a 180° rotation on the equator of the Bloch sphere about the polar, or “Z”, axis. (It is noteworthy in this regard that the 180° Z-rotation on the equator of the Block sphere corresponds to a two qubit SWAP operation between two single-electron spin-½ qubits, and a similar 90° rotation corresponds to a √{square root over (SWAP)} operation. These are fundamental operations of quantum logic that are well known to those skilled in the art of quantum computation.)
Thus, a suitably programmed voltage pulse applied to the gate electrodes can produce any desired rotation (i.e., a “Z-rotation”) of a suitably prepared initial state around the equator of the Bloch sphere.
All possible two-electron spin states of the double dot, i.e., all points on the surface of the Bloch sphere, could be reached by combining a Z-rotation with an X-rotation, i.e., with a rotation about an axis perpendicular to the Z axis, so that θ, rather than φ, is varied.
Researchers in the field have in fact proposed various approaches to two-axis control. For example, an approach for two-axis control of singlet-triplet qubits in GaAs/AlGaAs has been published in J.R. Petta et al., “Coherent manipulation of coupled electron spins in semiconductor quantum dots,” Science 309, 2180-2184 (2005). Approaches for two-axis control of singlet-triplet qubits in Si/SiGe have been published in B. M. Maune et al., “Coherent singlet-triplet oscillations in a silicon-based double quantum dot,” Nature 481, 344-347 (2012), and in X. Wu et al., “Two-axis control of a singlet-triplet qubit with an integrated micromagnet,” Proc. Natl Acad. Sci. USA 111, 11938-11942 (2014).
Approaches for two-axis control of single-spin (spin-½) qubits in GaAs/AlGaAs have been published in F. H. L. Koppens et al., “Driven coherent oscillations of a single electron spin in a quantum dot,” Nature (London) 442, 766-771 (2006), and in M. Pioro-Ladrière et al., “Electrically driven single-electron spin resonance in a slanting Zeeman field,” Nat. Phys. 4, 776-779 (2008).
An approach for two-axis control of single-spin (spin-½) qubits in Si/SiGe has been published in E. Kawakami et al., “Electrical control of a long-lived spin qubit in a Si/SiGe quantum dot,” Nat. Nano 9, 666-670 (2014).
Approaches for two-axis control of single-spin (spin-½) qubits in MOS have been published in M. Veldhorst et al., “An addressable quantum dot qubit with fault-tolerant control-fidelity,” Nat. Nano 9, 981-985 (2014), and in A. Corna et al., “Electrically driven electron spin resonance mediated by spin-valley-orbit coupling in a silicon quantum dot,” Nature PJ Quantum Information 4, Article number 6 (2018), (seven pages), www.nature.com/npjqi.
The entirety of each of the above-cited references is hereby incorporated herein by reference.
Although two-axis control is described in the above-cited references among others, the approaches that are known until now call for large on-chip microwave strip-lines, controlled hyperfine interactions with material nuclear spins, precisely-positioned nanomagnets, or specialized fin geometries for corner-QD effects.
If, on the other hand, two-axis control could be achieved through simple control of electrical voltages and currents, it would open up further possibilities for the design of quantum logic gates. Thus, there remains a need for simpler and more practical implementations that offer two-axis control of the spin state in lateral double dots.